Bottom Line:
We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view.In particular, we show that type I excitability is associated with a novel singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation.In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures.

ABSTRACTWe discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a novel singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures. We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.

Mentions:
Let us mention, without proof, that the homoclinic saddle-loop bifurcation curve ( in Fig. 10) of the Bogdanov–Takens point (BT) at lies between the Andronov–Hopf curve (AH) and the parameter line () and tends towards this parameter line as it approaches infinity; see Fig. 10. This can be seen by studying (33) for near infinity [22]. Fig. 10

Mentions:
Let us mention, without proof, that the homoclinic saddle-loop bifurcation curve ( in Fig. 10) of the Bogdanov–Takens point (BT) at lies between the Andronov–Hopf curve (AH) and the parameter line () and tends towards this parameter line as it approaches infinity; see Fig. 10. This can be seen by studying (33) for near infinity [22]. Fig. 10

Bottom Line:
We discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view.In particular, we show that type I excitability is associated with a novel singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation.In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures.

ABSTRACTWe discuss the notion of excitability in 2D slow/fast neural models from a geometric singular perturbation theory point of view. We focus on the inherent singular nature of slow/fast neural models and define excitability via singular bifurcations. In particular, we show that type I excitability is associated with a novel singular Bogdanov-Takens/SNIC bifurcation while type II excitability is associated with a singular Andronov-Hopf bifurcation. In both cases, canards play an important role in the understanding of the unfolding of these singular bifurcation structures. We also explain the transition between the two excitability types and highlight all bifurcations involved, thus providing a complete analysis of excitability based on geometric singular perturbation theory.